Problem 6 - College Algebra Essentials

Question

Write each equation in its equivalent exponential form. $$3=\log _{b} 27$$

Step-by-Step Solution

Answer

The exponential form of the equation \(3 = \log_b 27\) is \(b^3 = 27\).

Step 1: Understand the Logarithm to Exponential transformation

A logarithmic equation can be converted to exponential form by following this rule: If \(y = \log_b x\), then the equivalent expression in exponential form is given by \(b^y = x\).

Step 2: Substitute the given values

After understanding the rule, now substitute the values from the given logarithmic equation \(3 = \log_b 27\) into the rule. So, \(y\) is 3, \(b\) is yet unknown, and \(x\) is 27.

Step 3: Write the Exponential Form

Substituting the values obtain the exponential form of the given problem: \(b^3 = 27\).